Example Data Table
| Successes |
Sample Size |
p0 |
Alternative |
Z Score |
P Value |
Decision at 0.05 |
| 56 |
100 |
0.50 |
p > p0 |
1.2000 |
0.1151 |
Do not reject |
| 72 |
120 |
0.50 |
p ≠ p0 |
2.1909 |
0.0285 |
Reject |
| 18 |
80 |
0.30 |
p < p0 |
-1.4639 |
0.0716 |
Do not reject |
Formula Used
Sample proportion: p̂ = x / n
Null standard error: SE0 = √[p0(1 - p0) / n]
Z statistic: z = (p̂ - p0) / SE0
Right tailed p value: P(Z ≥ z)
Left tailed p value: P(Z ≤ z)
Two sided p value: 2 × P(Z ≥ |z|)
Confidence interval: p̂ ± z* × √[p̂(1 - p̂) / n]
How to Use This Calculator
- Enter the number of successes observed in your sample.
- Enter the full sample size.
- Enter the hypothesized population proportion.
- Select the correct alternative hypothesis.
- Set alpha and confidence level.
- Choose whether to apply continuity correction.
- Press the calculate button.
- Review the result, then export CSV or PDF if needed.
Understanding the One Proportion Z Test
A one proportion z test checks one sample rate against a claimed population rate. It is useful when the outcome has two states, such as yes or no, pass or fail, converted or not converted. The method compares the observed sample proportion with the hypothesized proportion. It then measures the gap in standard error units.
Why It Matters
This test helps researchers judge whether a sample result is likely by chance. A marketing team may test a signup rate. A factory may test a defect rate. A health analyst may test a response rate. Each case needs a clear null value and a suitable sample size.
Good Input Choices
The sample size should be large enough for normal approximation. A common rule checks expected successes and expected failures under the null. Both should usually be at least five. Use exact methods when counts are small or the population process is highly uneven.
Interpreting Results
The z score shows direction and distance. A positive value means the sample proportion is above the null proportion. A negative value means it is below it. The p value reports the probability of seeing evidence this extreme, assuming the null claim is true. A small p value supports rejecting the null.
Confidence Interval Use
A confidence interval estimates a plausible range for the true population proportion. This calculator uses the selected confidence level and sample standard error for the interval. If the null value falls far outside the interval, the test result often agrees with rejection.
Careful Reporting
Report the sample size, successes, sample proportion, null proportion, alternative hypothesis, z score, p value, and decision. Mention whether continuity correction was used. Avoid saying the null is proven true. A non significant result only means the sample did not provide enough evidence.
Example Uses
Use this tool for surveys, trials, checks, and website tests. Keep the question simple. Define success before entering data. Save exported files for review later.
Practical Limits
The test assumes random sampling and independent observations. Biased sampling can make a precise looking result wrong. Very large samples can make tiny differences significant. Always compare statistical significance with practical importance. Good decisions need both numbers and context.
FAQs
What is a population proportion z test?
It is a hypothesis test for one sample proportion. It compares your observed sample rate with a claimed population rate and returns a z score, p value, confidence interval, and decision.
When should I use this calculator?
Use it when your data has two outcomes, such as success or failure. The sample should be random, observations should be independent, and expected successes and failures should usually be at least five.
What does p0 mean?
p0 is the claimed population proportion under the null hypothesis. For example, if the claim says the true conversion rate is 40%, enter 0.40 or 40.
What does the z score show?
The z score shows how far the sample proportion is from the hypothesized proportion. It measures that distance in null standard error units.
What is a small p value?
A small p value means the sample result is unlikely if the null claim is true. If the p value is less than or equal to alpha, reject the null hypothesis.
Should I use continuity correction?
Continuity correction adjusts the discrete count before using the normal curve. It can be useful for moderate sample sizes, but it may make the test more conservative.
Can I enter percentages?
Yes. You may enter proportions as decimals or percentages. For example, 0.25 and 25 both represent twenty five percent.
What should I report from the result?
Report successes, sample size, sample proportion, p0, alternative hypothesis, z score, p value, confidence interval, alpha, and the final decision.